1. The document introduces parameters like mean, variance, and covariance to represent the relationship between two random variables X and Y.
2. Covariance is defined as a measure of how X and Y vary together, while correlation is the standardized covariance between X and Y.
3. Two random variables are uncorrelated if their covariance is 0, while independence implies uncorrelatedness but not vice versa, as shown by an example of dependent but uncorrelated random variables Z and W.
Joint moments and joint characteristic functionsNebiyu Mohammed
- The document introduces joint moments and joint characteristic functions to compactly represent the information contained in the joint probability density function (p.d.f.) of two random variables (RVs).
- It defines covariance as a measure of how two RVs vary in relation to each other. Covariance and correlation coefficient are introduced to represent the cross-behavior between two RVs, analogous to how mean and variance represent the behavior of a single RV.
- It shows that independence implies uncorrelatedness between RVs, but the converse is not necessarily true. The joint characteristic function is defined and useful properties are derived for computing moments and representing Gaussian RVs.
Two Random variable based on probability lect7a.pptsadafshahbaz7777
1) The document discusses joint probability distributions of two random variables X and Y. It defines their joint probability distribution function fXY(x,y) and describes some of its key properties.
2) It also discusses how to calculate marginal probabilities and marginal probability distribution functions from the joint distribution. The marginal distributions of X and Y can be obtained by summing/integrating the joint distribution over one of the variables.
3) Examples are provided to demonstrate calculating probabilities of events involving X and Y, as well as marginal distributions, from a given joint distribution.
Mean, variable , moment characteris function.pptsadafshahbaz7777
The document introduces the concepts of mean and variance, which are universally used to characterize the average behavior and overall properties of a random variable and its probability density function. The mean represents the average value of the random variable, while the variance represents the average squared deviation from the mean and indicates how concentrated or spread out the random variable is around the mean. Formulas are provided to calculate the mean and variance for various common distributions such as normal, binomial, Poisson, and exponential distributions. The parameter of each distribution is shown to represent the mean value in many cases.
This document discusses mean square estimation and the minimization of mean square error (MMSE) criterion for obtaining estimators. It shows that under MMSE, the best estimator for an unknown quantity Y in terms of observed data X is the conditional mean of Y given X. This minimizes the mean square error between the estimate and the true value of Y. For linear estimators, the best estimator satisfies an orthogonality principle where the error is orthogonal to the data. A similar nonlinear orthogonality rule also holds for nonlinear estimators.
The document discusses conditional probability density functions and conditional expected values. It defines the conditional probability density function of a random variable X given another random variable Y=y as f(x|y). This allows updating knowledge about X based on information about Y. The conditional expected value of X given Y=y is defined as the integral of xf(x|y)dx. An example calculates the conditional PDFs and expected value for two random variables with a given joint PDF.
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
The document discusses determining the probability density function (PDF) of a new random variable Z that is defined as a function g(X,Y) of two random variables X and Y. It provides examples of finding the PDF of Z=X+Y when X and Y are independent random variables with various distributions such as exponential or uniform. The key points are:
1) The PDF of Z, fZ(z), can be found by evaluating the integral of the joint PDF of X and Y, fXY(x,y), over the region where g(x,y) ≤ z.
2) If X and Y are independent, fZ(z) is the convolution of the PDFs of X
Chapter 6 simple regression and correlationRione Drevale
There is a significant positive correlation between amount of feed intake and live weight of broilers. The correlation coefficient (r) between feed intake and live weight is 0.726, which is statistically significant with p<0.017. On average, broilers gain approximately 0.5 kg of live weight for every 1 kg of feed consumed.
Joint moments and joint characteristic functionsNebiyu Mohammed
- The document introduces joint moments and joint characteristic functions to compactly represent the information contained in the joint probability density function (p.d.f.) of two random variables (RVs).
- It defines covariance as a measure of how two RVs vary in relation to each other. Covariance and correlation coefficient are introduced to represent the cross-behavior between two RVs, analogous to how mean and variance represent the behavior of a single RV.
- It shows that independence implies uncorrelatedness between RVs, but the converse is not necessarily true. The joint characteristic function is defined and useful properties are derived for computing moments and representing Gaussian RVs.
Two Random variable based on probability lect7a.pptsadafshahbaz7777
1) The document discusses joint probability distributions of two random variables X and Y. It defines their joint probability distribution function fXY(x,y) and describes some of its key properties.
2) It also discusses how to calculate marginal probabilities and marginal probability distribution functions from the joint distribution. The marginal distributions of X and Y can be obtained by summing/integrating the joint distribution over one of the variables.
3) Examples are provided to demonstrate calculating probabilities of events involving X and Y, as well as marginal distributions, from a given joint distribution.
Mean, variable , moment characteris function.pptsadafshahbaz7777
The document introduces the concepts of mean and variance, which are universally used to characterize the average behavior and overall properties of a random variable and its probability density function. The mean represents the average value of the random variable, while the variance represents the average squared deviation from the mean and indicates how concentrated or spread out the random variable is around the mean. Formulas are provided to calculate the mean and variance for various common distributions such as normal, binomial, Poisson, and exponential distributions. The parameter of each distribution is shown to represent the mean value in many cases.
This document discusses mean square estimation and the minimization of mean square error (MMSE) criterion for obtaining estimators. It shows that under MMSE, the best estimator for an unknown quantity Y in terms of observed data X is the conditional mean of Y given X. This minimizes the mean square error between the estimate and the true value of Y. For linear estimators, the best estimator satisfies an orthogonality principle where the error is orthogonal to the data. A similar nonlinear orthogonality rule also holds for nonlinear estimators.
The document discusses conditional probability density functions and conditional expected values. It defines the conditional probability density function of a random variable X given another random variable Y=y as f(x|y). This allows updating knowledge about X based on information about Y. The conditional expected value of X given Y=y is defined as the integral of xf(x|y)dx. An example calculates the conditional PDFs and expected value for two random variables with a given joint PDF.
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
The document discusses determining the probability density function (PDF) of a new random variable Z that is defined as a function g(X,Y) of two random variables X and Y. It provides examples of finding the PDF of Z=X+Y when X and Y are independent random variables with various distributions such as exponential or uniform. The key points are:
1) The PDF of Z, fZ(z), can be found by evaluating the integral of the joint PDF of X and Y, fXY(x,y), over the region where g(x,y) ≤ z.
2) If X and Y are independent, fZ(z) is the convolution of the PDFs of X
Chapter 6 simple regression and correlationRione Drevale
There is a significant positive correlation between amount of feed intake and live weight of broilers. The correlation coefficient (r) between feed intake and live weight is 0.726, which is statistically significant with p<0.017. On average, broilers gain approximately 0.5 kg of live weight for every 1 kg of feed consumed.
This document discusses the geometry of an unified contact Riemannian manifold equipped with a semi-symmetric metric S-connection. It derives the curvature tensor R of the manifold relative to this connection. It shows that if the manifold admits such a connection whose curvature tensor is locally isometric to the unit sphere, then the conformal and con-harmonic curvature tensors coincide if a certain condition holds. It also shows that in this case, the con-circular curvature tensor coincides with the Riemannian connection under the same condition. Some other geometrically important results and theorems are obtained.
This document discusses functions of random variables. It defines a random variable Y as a function g(X) of another random variable X. If g(x) is a Borel function, then Y will also be a random variable. The probability distribution and density functions of Y can be determined based on those of X. Several examples are provided to illustrate these concepts, including linear, absolute value, half-wave rectifier, and other functions of a random variable X.
Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing func...IOSR Journals
The document presents theorems characterizing when the Hardy-Steklov operator is bounded from one two-exponent Lorentz space to another. Specifically, it provides conditions on weights v and w such that the operator is bounded from L(0,∞)qpv to L(0,∞)srw. It defines the Hardy-Steklov operator and two-exponent Lorentz spaces. It states two theorems that characterize the weights using inequalities involving the weights and derivatives of the functions defining the Hardy-Steklov operator. The theorems assume the functions satisfy certain conditions like being strictly increasing and having derivatives satisfying an inequality.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Random variable, distributive function lect3a.pptsadafshahbaz7777
1. A random variable (X) is a function that maps outcomes of a probability experiment to real numbers. The probability distribution function (FX(x)) gives the probability that X takes on a value less than or equal to x.
2. FX(x) satisfies properties of a distribution function - it is nondecreasing, right-continuous, and its limit as x approaches positive/negative infinity is 0/1.
3. A random variable can be either continuous or discrete. FX(x) is continuous if it is continuous for all x, and discrete if it has jump discontinuities at countable points.
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
Probability and statistics - Discrete Random Variables and Probability Distri...Asma CHERIF
1. A random variable is discrete if the probabilities of it being equal to each value sum to 1. This means its distribution can be described by listing the probabilities of each possible value.
2. For a discrete random variable X, its probability function pX(x) gives the probability P(X=x) of X being equal to each value x. This fully describes the distribution of X.
3. The law of total probability states that for a partition of outcomes into events A1, A2, etc., the probability of event B is the sum of the probabilities of B given each Ai, weighted by the probability of Ai.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
I am Ahmed M. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from New York University, Abu Dhabi. I have been helping students with their assignments for the past 10 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems Assignment.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
- Covariance and correlation are measures used to summarize the relationship between two random variables with a joint probability distribution.
- Covariance measures the degree of linear relationship between the variables, ranging from -σXσY to +σXσY, where a value of 0 indicates either independence or a nonlinear relationship.
- Correlation is the covariance normalized by the product of the standard deviations, ranging from -1 to 1, providing a dimensionless measure of the strength and type (positive or negative) of linear dependence.
This document describes a proposed implementation of the Region Connection Calculus (RCC) using constraint handling rules (CHR). It defines the 8 base RCC relations in terms of connectedness relations. The implementation uses CHR to define simplification, propagation, and simpagation rules to represent and reason about RCC constraints. Some key relations like connection, disconnection, and part are defined through corresponding CHR rules based on their definitions in RCC axioms. The implementation aims to simplify constraints into just connections but has some limitations, such as not fully capturing part relations due to inability to define global constraints.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
The document defines and provides theory on triple integrals. It discusses that triple integrals are used to calculate volumes or integrate over a fourth dimension based on three other dimensions. Examples are provided on setting up and evaluating triple integrals in rectangular, cylindrical, and polar coordinates. Specific steps are outlined for solving triple integrals over different shapes and order of integration.
This document discusses the geometry of an unified contact Riemannian manifold equipped with a semi-symmetric metric S-connection. It derives the curvature tensor R of the manifold relative to this connection. It shows that if the manifold admits such a connection whose curvature tensor is locally isometric to the unit sphere, then the conformal and con-harmonic curvature tensors coincide if a certain condition holds. It also shows that in this case, the con-circular curvature tensor coincides with the Riemannian connection under the same condition. Some other geometrically important results and theorems are obtained.
This document discusses functions of random variables. It defines a random variable Y as a function g(X) of another random variable X. If g(x) is a Borel function, then Y will also be a random variable. The probability distribution and density functions of Y can be determined based on those of X. Several examples are provided to illustrate these concepts, including linear, absolute value, half-wave rectifier, and other functions of a random variable X.
Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing func...IOSR Journals
The document presents theorems characterizing when the Hardy-Steklov operator is bounded from one two-exponent Lorentz space to another. Specifically, it provides conditions on weights v and w such that the operator is bounded from L(0,∞)qpv to L(0,∞)srw. It defines the Hardy-Steklov operator and two-exponent Lorentz spaces. It states two theorems that characterize the weights using inequalities involving the weights and derivatives of the functions defining the Hardy-Steklov operator. The theorems assume the functions satisfy certain conditions like being strictly increasing and having derivatives satisfying an inequality.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Random variable, distributive function lect3a.pptsadafshahbaz7777
1. A random variable (X) is a function that maps outcomes of a probability experiment to real numbers. The probability distribution function (FX(x)) gives the probability that X takes on a value less than or equal to x.
2. FX(x) satisfies properties of a distribution function - it is nondecreasing, right-continuous, and its limit as x approaches positive/negative infinity is 0/1.
3. A random variable can be either continuous or discrete. FX(x) is continuous if it is continuous for all x, and discrete if it has jump discontinuities at countable points.
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
Probability and statistics - Discrete Random Variables and Probability Distri...Asma CHERIF
1. A random variable is discrete if the probabilities of it being equal to each value sum to 1. This means its distribution can be described by listing the probabilities of each possible value.
2. For a discrete random variable X, its probability function pX(x) gives the probability P(X=x) of X being equal to each value x. This fully describes the distribution of X.
3. The law of total probability states that for a partition of outcomes into events A1, A2, etc., the probability of event B is the sum of the probabilities of B given each Ai, weighted by the probability of Ai.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
I am Ahmed M. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from New York University, Abu Dhabi. I have been helping students with their assignments for the past 10 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems Assignment.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
- Covariance and correlation are measures used to summarize the relationship between two random variables with a joint probability distribution.
- Covariance measures the degree of linear relationship between the variables, ranging from -σXσY to +σXσY, where a value of 0 indicates either independence or a nonlinear relationship.
- Correlation is the covariance normalized by the product of the standard deviations, ranging from -1 to 1, providing a dimensionless measure of the strength and type (positive or negative) of linear dependence.
This document describes a proposed implementation of the Region Connection Calculus (RCC) using constraint handling rules (CHR). It defines the 8 base RCC relations in terms of connectedness relations. The implementation uses CHR to define simplification, propagation, and simpagation rules to represent and reason about RCC constraints. Some key relations like connection, disconnection, and part are defined through corresponding CHR rules based on their definitions in RCC axioms. The implementation aims to simplify constraints into just connections but has some limitations, such as not fully capturing part relations due to inability to define global constraints.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
The document defines and provides theory on triple integrals. It discusses that triple integrals are used to calculate volumes or integrate over a fourth dimension based on three other dimensions. Examples are provided on setting up and evaluating triple integrals in rectangular, cylindrical, and polar coordinates. Specific steps are outlined for solving triple integrals over different shapes and order of integration.
The Sufis played a key role in spreading Islam in the Indian subcontinent through their missionary work, exemplary character, and humanitarian activities. They established Sufi orders and fraternities that attracted many converts to Islam. The Sufis emphasized spiritual worship, simplicity, and equality, which appealed to many in India's rigid caste system. Their khanqahs, or centers, provided spiritual guidance, food for the poor, and promoted religious harmony. Major Sufi orders like the Chishtiyah and Suhrawardiyah helped establish Islam throughout South Asia.
This document provides an overview of diasporas, including conceptualizing the term, important characteristics, typologies, the importance of diasporas, and challenges. It specifically examines the Chinese and Indian diasporas in detail, comparing their histories, distributions, cultures, and experiences. Key points covered include the large sizes and wide distributions of the Chinese and Indian diasporas, with the Chinese diaspora being mostly in Southeast Asia and the Indian diaspora more widely scattered. State responses to diasporas and issues around citizenship and identity are also discussed.
This document provides an introduction to social studies as a field of study. It defines social studies as the multidisciplinary study of past, present, and future societies from cultural, economic, geographic, and political perspectives. The goals of social studies education are to help students understand their role in the world and develop critical thinking skills to participate competently as citizens. An effective social studies curriculum incorporates 10 themes including culture, time and change, individuals and groups, power and governance, and global connections. Principles of effective social studies teaching are that it be meaningful, integrative, value-based, challenging, and active. The document also discusses reasons for lack of student interest in social studies, including an emphasis on other subjects and ineffective teaching
This document provides an overview of Dimension 3 of the C3 Framework, which focuses on evaluating sources and using evidence. It discusses two main indicators: gathering and evaluating sources, and developing claims and using evidence. For gathering and evaluating sources, it emphasizes finding information from various sources and determining relevance. For developing claims, it stresses the ability to understand relationships between claims and evidence and to select evidence purposefully to support arguments. The document also introduces the SOURCES framework for evaluating sources and provides examples of how to guide students through each step, from scrutinizing the fundamental source to summarizing final thoughts. It directs teachers to Library of Congress resources for primary sources.
Pakistan has had 4 constitutions since its independence in 1947. The interim constitution was the Government of India Act of 1935. The constitutions of 1956, 1962, and 1973 were later adopted but all faced amendments or abrogation. The 1973 constitution remains in effect today but has faced numerous amendments over controversial issues like the powers of the Prime Minister and President. Pakistan's constitutional development has been impacted by political instability, martial laws, the language issue, and the distribution of powers between federal and provincial governments.
The major constitutional issues faced by Pakistan's first Constituent Assembly included federalism, representation, the national language, and defining the relationship between Islam and the state. There was debate around the division of powers between the central and provincial governments, representation in parliament given regional demographic differences, and whether to adopt Urdu or Bengali as the national language. The Assembly also discussed a parliamentary vs. presidential system and whether to define Pakistan as an Islamic or secular state. It ultimately passed the Objectives Resolution which established Pakistan's basic objectives and rejected theocracy while affirming Islam as the guiding principle.
This document discusses the challenges Pakistan faced in drafting its first constitution after independence in 1947. There was debate around whether Pakistan should be a secular democratic state or an Islamic state governed by Islamic law. Supporters of an Islamic state, known as the Ulema, wanted a system that closely followed 7th century Islamic principles, while others advocated for a modern democratic system that respected Islamic values. This debate led to delays in the constitution drafting process and disagreements over the role of Islamic law. The proposed constitution attempted to balance these views by including some references to Islam but largely establishing a secular democratic system with protections for religious minorities.
The document provides information about Adverse Childhood Experiences (ACEs) including:
1) It summarizes the objectives of raising awareness about ACE research and prevention frameworks like Essentials for Childhood.
2) It describes the original ACE study which found associations between childhood trauma and later health outcomes, and notes that 64% of participants experienced at least one ACE.
3) It shows that higher ACE scores correlate with increased risks of health problems like smoking, alcoholism, suicide attempts, and teen pregnancy.
4) Data from the Kansas BRFSS survey found rates of ACEs in Kansas similar to national data, with certain groups like women and low-income residents
This document provides an overview of introductory physics course sequences at Columbia University for engineering and science students. There are three main sequences - the 1400 sequence emphasizes basics and problem solving, the 1600 sequence is more mathematical and theoretical, and the 2800 sequence condenses the 1600 material into two semesters for advanced students. The document outlines topics, textbooks, recommended grades, and separate lab courses for each sequence. It also previews the upcoming material and chapters to be covered in Physics 1401, including one-dimensional motion, vectors, and scales in physics.
This document discusses key concepts in financial management including business units, types of business organizations, assets, basic financial problems, the role of financial managers and markets, and goals of financial management. It also covers topics like productivity and profitability as scale increases, the effects of financial leverage based on cost of debt and debt-equity ratios, and how these factors can impact investment profitability.
This document provides information about a biochemistry course at the University of Houston, including the instructor's contact information, office hours, and research interests in molecular cell adhesion mechanisms using nuclear magnetic resonance spectroscopy. It summarizes the topics covered in the first lecture, including the origins of life from simple organic molecules and the evolution of prokaryotic and eukaryotic cells. The document also previews the topics to be covered in the next lecture on thermodynamics and chemical equilibria.
This document provides an overview of a symposium on culturally effective care for LGBT populations. It begins with an agenda that includes differentiating key terms, defining intersectionality, identifying health disparities and social determinants of health, and applying concepts through a case study. The document then defines various terms related to gender identity, sexual orientation, sex, and development. It reviews the history of pathologization of LGBT identities in medicine and mental health. Statistics on demographics and health disparities experienced by LGBT populations are presented. Strategies for providing culturally effective care include creating an inclusive environment, building trust, ensuring confidentiality, and using inclusive language. Local and national resources for LGBT care are also listed.
Counter Terrorism Department Jobs 2024 | CTD Jobs in Punjab PoliceMerrie rp
Counter Terrorism Department Jobs in Punjab Police are announced through Punjab Public Service Commission. The details of Jobs in Punjab Police Counter Terrorism Department is given below:
Carporal (BS-11)
TOTAL POSTS:467
AGE:
Male:
18 to 25
Female:
18 to 25 years
Age & Sex of the Transgender will be based on the contents of their CNIC
GENDER:
Male, Female
DOMICILE:
All Punjab Basis
PLACE OF POSTING:
Anywhere in Punjab
SYLLABUS FOR WRITTEN EXAMINATION/ TEST (IF HELD)
One Paper MCQs Type Written Test of 100 marks
(90 minutes duration) comprising following
subjects:
a) General Knowledge, Pakistan Studies, Current Affairs, Geography. Questions related to Counter Terrorism Department and its functions, NACTA, FATF, Terrorism, National Action Plan and Basics of Anti-Terrorism Laws in Pakistan.
b) English language comprehension including Synonyms, Antonyms, Sentence Correction/ Completion, One word substitution and idioms.
c) Usage of Basic Softwares like M.S Office, Electronic Record Keeping, Internet, E-mail etc.
LinkedIn Strategic Guidelines for June 2024Bruce Bennett
LinkedIn is a powerful tool for networking, researching, and marketing yourself to clients and employers. This session teaches strategic practices for building your LinkedIn internet presence and marketing yourself. The use of # and @ symbols is covered as well as going mobile with the LinkedIn app.
Delta International is an ISO Certified top recruiting agency in Pakistan, recognized for its highly experienced recruiters. With a diverse range of international jobs for Pakistani workers, Delta International maintains extensive connections with overseas employers, making it one of the top 10 recruitment agencies in Pakistan. It stands out in the list of recruitment agencies in Pakistan for its exceptional services.
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Joint moment and joint character lectr10a.ppt
1. 1
10. Joint Moments and Joint Characteristic
Functions
Following section 6, in this section we shall introduce
various parameters to compactly represent the information
contained in the joint p.d.f of two r.vs. Given two r.vs X and
Y and a function define the r.v
Using (6-2), we can define the mean of Z to be
(10-1)
)
,
( Y
X
g
Z
),
,
( y
x
g
.
)
(
)
(
dz
z
f
z
Z
E Z
Z
(10-2)
PILLAI
2. 2
However, the situation here is similar to that in (6-13), and
it is possible to express the mean of in terms
of without computing To see this, recall from
(5-26) and (7-10) that
where is the region in xy plane satisfying the above
inequality. From (10-3), we get
As covers the entire z axis, the corresponding regions
are nonoverlapping, and they cover the entire xy plane.
z
D
y
x
XY
Z
y
x
y
x
f
z
z
Y
X
g
z
P
z
z
f
z
z
Z
z
P
)
,
(
)
,
(
)
,
(
)
(
(10-3)
z
D
)
,
( Y
X
g
Z
)
,
( y
x
fXY ).
(z
fZ
( , )
( ) ( , ) ( , ) .
z
Z XY
x y D
z f z z g x y f x y x y
(10-4)
z
D
z
PILLAI
3. 3
By integrating (10-4), we obtain the useful formula
or
If X and Y are discrete-type r.vs, then
Since expectation is a linear operator, we also get
( ) ( ) ( , ) ( , ) .
Z XY
E Z z f z dz g x y f x y dxdy
(10-5)
(10-6)
[ ( , )] ( , ) ( , ) .
XY
E g X Y g x y f x y dxdy
[ ( , )] ( , ) ( , ).
i j i j
i j
E g X Y g x y P X x Y y
(10-7)
( , ) [ ( , )].
k k k k
k k
E a g X Y a E g X Y
(10-8)
PILLAI
4. 4
If X and Y are independent r.vs, it is easy to see that
and are always independent of each other. In that
case using (10-7), we get the interesting result
However (10-9) is in general not true (if X and Y are not
independent).
In the case of one random variable (see (10- 6)), we defined
the parameters mean and variance to represent its average
behavior. How does one parametrically represent similar
cross-behavior between two random variables? Towards
this, we can generalize the variance definition given in
(6-16) as shown below:
)
(X
g
Z
)].
(
[
)]
(
[
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)]
(
)
(
[
Y
h
E
X
g
E
dy
y
f
y
h
dx
x
f
x
g
dxdy
y
f
x
f
y
h
x
g
Y
h
X
g
E
Y
X
Y
X
(10-9)
)
(Y
h
W
PILLAI
5. 5
Covariance: Given any two r.vs X and Y, define
By expanding and simplifying the right side of (10-10), we
also get
It is easy to see that
To see (10-12), let so that
.
)
(
)
(
)
(
)
(
)
,
(
__
__
____
Y
X
XY
Y
E
X
E
XY
E
XY
E
Y
X
Cov Y
X
(10-10)
(10-12)
.
)
(
)
(
)
,
( Y
X Y
X
E
Y
X
Cov
(10-11)
,
Y
aX
U
.
)
(
)
(
)
,
( Y
Var
X
Var
Y
X
Cov
.
0
)
(
)
,
(
2
)
(
)
(
)
(
)
(
2
2
Y
Var
Y
X
Cov
a
X
Var
a
Y
X
a
E
U
Var Y
X
(10-13)
PILLAI
6. 6
The right side of (10-13) represents a quadratic in the
variable a that has no distinct real roots (Fig. 10.1). Thus the
roots are imaginary (or double) and hence the discriminant
must be non-positive, and that gives (10-12). Using (10-12),
we may define the normalized parameter
or
and it represents the correlation
coefficient between X and Y.
(10-14)
)
(
)
(
)
,
(
2
Y
Var
X
Var
Y
X
Cov
,
1
1
,
)
,
(
)
(
)
(
)
,
(
XY
Y
X
XY
Y
X
Cov
Y
Var
X
Var
Y
X
Cov
a
)
(U
Var
Fig. 10.1
Y
X
XY
Y
X
Cov
)
,
( (10-15)
PILLAI
7. 7
Uncorrelated r.vs: If then X and Y are said to be
uncorrelated r.vs. From (11), if X and Y are uncorrelated,
then
Orthogonality: X and Y are said to be orthogonal if
From (10-16) - (10-17), if either X or Y has zero mean, then
orthogonality implies uncorrelatedness also and vice-versa.
Suppose X and Y are independent r.vs. Then from (10-9)
with we get
and together with (10-16), we conclude that the random
variables are uncorrelated, thus justifying the original
definition in (10-10). Thus independence implies
uncorrelatedness.
(10-16)
,
0
XY
,
)
(
,
)
( Y
Y
h
X
X
g
).
(
)
(
)
( Y
E
X
E
XY
E
.
0
)
(
XY
E
(10-17)
),
(
)
(
)
( Y
E
X
E
XY
E
PILLAI
8. 8
Naturally, if two random variables are statistically
independent, then there cannot be any correlation between
them However, the converse is in general not
true. As the next example shows, random variables can be
uncorrelated without being independent.
Example 10.1: Let Suppose X and Y
are independent. Define Z = X + Y, W = X - Y . Show that Z
and W are dependent, but uncorrelated r.vs.
Solution: gives the only solution set to be
Moreover
and
),
1
,
0
(
U
X ).
1
,
0
(
U
Y
|
|
,
2
,
2
,
1
1
,
2
0 w
z
w
z
w
z
w
z
.
2
/
1
|
)
,
(
|
w
z
J
.
2
,
2
w
z
y
w
z
x
y
x
w
y
x
z
,
).
0
(
XY
PILLAI
10. 10
and
Clearly Thus Z and W are not
independent. However
and
and hence
implying that Z and W are uncorrelated random variables.
,
otherwise
,
0
,
2
1
,
2
,
1
0
,
)
(
)
(
)
( z
z
z
z
z
f
z
f
z
f Y
X
Z
(10-19)
(10-20)
).
(
)
(
)
,
( w
f
z
f
w
z
f W
Z
ZW
(10-21)
,
0
)
(
)
(
)
)(
(
)
( 2
2
Y
E
X
E
Y
X
Y
X
E
ZW
E
0
)
(
)
(
)
(
)
,
(
W
E
Z
E
ZW
E
W
Z
Cov (10-22)
.
otherwise
,
0
,
1
1
|,
|
1
2
1
)
,
(
)
(
|
|
2 w
w
dz
dz
w
z
f
w
f
w
|w|
ZW
W
,
0
)
(
)
(
Y
X
E
W
E
PILLAI
11. 11
Example 10.2: Let Determine the variance of Z
in terms of and
Solution:
and using (10-15)
In particular if X and Y are independent, then and
(10-23) reduces to
Thus the variance of the sum of independent r.vs is the sum
of their variances
(10-23)
,
0
XY
.
bY
aX
Z
Y
X
, .
XY
( ) ( )
Z X Y
E Z E aX bY a b
2
2 2
2 2 2 2
2 2 2 2
( ) ( ) ( ) ( )
( ) 2 ( )( ) ( )
2 .
Z Z X Y
X X Y Y
X XY X Y Y
Var Z E Z E a X b Y
a E X abE X Y b E Y
a ab b
.
2
2
2
2
2
Y
X
Z b
a
(10-24)
).
1
(
b
a PILLAI
12. 12
Moments:
represents the joint moment of order (k,m) for X and Y.
Following the one random variable case, we can define the
joint characteristic function between two random variables
which will turn out to be useful for moment calculations.
Joint characteristic functions:
The joint characteristic function between X and Y is defined
as
Note that
,
)
,
(
]
[
dy
dx
y
x
f
y
x
Y
X
E XY
m
k
m
k
(10-25)
( ) ( )
( , ) ( , ) .
j Xu Yv j Xu Yv
XY XY
u v E e e f x y dxdy
(10-26)
.
1
)
0
,
0
(
)
,
(
XY
XY v
u
PILLAI
13. 13
It is easy to show that
If X and Y are independent r.vs, then from (10-26), we obtain
Also
More on Gaussian r.vs :
From Lecture 7, X and Y are said to be jointly Gaussian
as if their joint p.d.f has the form in (7-
23). In that case, by direct substitution and simplification,
we obtain the joint characteristic function of two jointly
Gaussian r.vs to be
.
)
,
(
1
)
(
0
,
0
2
2
v
u
XY
v
u
v
u
j
XY
E (10-27)
).
(
)
(
)
(
)
(
)
,
( v
u
e
E
e
E
v
u Y
X
jvY
juX
XY
(10-28)
( ) ( ,0), ( ) (0, ).
X XY Y XY
u u v v
(10-29)
),
,
,
,
,
( 2
2
Y
X
Y
X
N
PILLAI
14. 14
.
)
(
)
,
(
)
2
(
2
1
)
(
)
(
2
2
2
2
v
uv
u
v
u
j
Yv
Xu
j
XY
Y
Y
X
X
Y
X
e
e
E
v
u
(10-30)
Equation (10-14) can be used to make various conclusions.
Letting in (10-30), we get
and it agrees with (6-47).
From (7-23) by direct computation using (10-11), it is easy
to show that for two jointly Gaussian random variables
Hence from (10-14), in represents
the actual correlation coefficient of the two jointly Gaussian
r.vs in (7-23). Notice that implies
0
v
,
)
0
,
(
)
(
2
2
2
1
u
u
j
XY
X
X
X
e
u
u
(10-31)
)
,
,
,
,
( 2
2
Y
X
Y
X
N
.
)
,
( Y
X
Y
X
Cov
0
PILLAI
15. 15
Thus if X and Y are jointly Gaussian, uncorrelatedness does
imply independence between the two random variables.
Gaussian case is the only exception where the two concepts
imply each other.
Example 10.3: Let X and Y be jointly Gaussian r.vs with
parameters Define
Determine
Solution: In this case we can make use of characteristic
function to solve this problem.
).
,
,
,
,
( 2
2
Y
X
Y
X
N .
bY
aX
Z
).
(z
fZ
).
,
(
)
(
)
(
)
(
)
( )
(
bu
au
e
E
e
E
e
E
u
XY
jbuY
jauX
u
bY
aX
j
jZu
Z
(10-32)
).
(
)
(
)
,
( y
f
x
f
Y
X
f Y
X
XY
PILLAI
16. 16
From (10-30) with u and v replaced by au and bu
respectively we get
where
Notice that (10-33) has the same form as (10-31), and hence
we conclude that is also Gaussian with mean and
variance as in (10-34) - (10-35), which also agrees with (10-
23).
From the previous example, we conclude that any linear
combination of jointly Gaussian r.vs generate a Gaussian r.v.
,
)
(
2
2
2
2
2
2
2
2
1
)
2
(
2
1
)
( u
u
j
u
b
ab
a
u
b
a
j
Z
Z
Z
Y
Y
X
X
Y
X
e
e
u
(10-33)
(10-34)
(10-35)
bY
aX
Z
.
2
,
2
2
2
2
2
Y
Y
X
X
Z
Y
X
Z
b
ab
a
b
a
PILLAI
17. 17
In other words, linearity preserves Gaussianity. We can use
the characteristic function relation to conclude an even more
general result.
Example 10.4: Suppose X and Y are jointly Gaussian r.vs as
in the previous example. Define two linear combinations
what can we say about their joint distribution?
Solution: The characteristic function of Z and W is given by
As before substituting (10-30) into (10-37) with u and v
replaced by au + cv and bu + dv respectively, we get
.
, dY
cX
W
bY
aX
Z
(10-36)
).
,
(
)
(
)
(
)
(
)
,
(
)
(
)
(
)
(
)
(
)
(
dv
bu
cv
au
e
E
e
E
e
E
v
u
XY
dv
bu
jY
cv
au
jX
v
dY
cX
j
u
bY
aX
j
Wv
Zu
j
ZW
(10-37)
PILLAI
18. 18
,
)
,
(
)
2
(
2
1
)
( 2
2
2
2
v
uv
u
v
u
j
ZW
W
Y
X
ZW
Z
W
Z
e
v
u
(10-38)
where
and
From (10-38), we conclude that Z and W are also jointly
distributed Gaussian r.vs with means, variances and
correlation coefficient as in (10-39) - (10-43).
,
2
,
2
,
,
2
2
2
2
2
2
2
2
2
2
Y
Y
X
X
W
Y
Y
X
X
Z
Y
X
W
Y
X
Z
d
cd
c
b
ab
a
d
c
b
a
(10-39)
(10-40)
(10-41)
(10-42)
.
)
( 2
2
W
Z
Y
Y
X
X
ZW
bd
bc
ad
ac
(10-43)
PILLAI
19. 19
To summarize, any two linear combinations of jointly
Gaussian random variables (independent or dependent) are
also jointly Gaussian r.vs.
Of course, we could have reached the same conclusion by
deriving the joint p.d.f using the technique
developed in section 9 (refer (7-29)).
Gaussian random variables are also interesting because of
the following result:
Central Limit Theorem: Suppose are a set of
zero mean independent, identically distributed (i.i.d) random
Linear
operator
Gaussian input Gaussian output
)
,
( w
z
fZW
n
X
X
X ,
,
, 2
1
Fig. 10.3
PILLAI
20. 20
variables with some common distribution. Consider their
scaled sum
Then asymptotically (as )
Proof: Although the theorem is true under even more
general conditions, we shall prove it here under the
independence assumption. Let represent their common
variance. Since
we have
.
2
1
n
X
X
X
Y n
(10-44)
n
).
,
0
( 2
N
Y
2
,
0
)
(
i
X
E
.
)
(
)
( 2
2
i
i X
E
X
Var
(10-45)
(10-46)
(10-47)
PILLAI
21. 21
Consider
where we have made use of the independence of the
r.vs But
where we have made use of (10-46) - (10-47). Substituting
(10-49) into (10-48), we obtain
and as
n
i
X
n
i
n
u
jX
n
u
X
X
X
j
jYu
Y
n
u
e
E
e
E
e
E
u
i
i
n
1
1
/
/
)
(
)
/
(
)
(
)
(
)
( 2
1
(10-48)
.
,
,
, 2
1 n
X
X
X
,
1
2
1
!
3
!
2
1
)
( 2
/
3
2
2
2
/
3
3
3
3
2
2
2
/
n
o
n
u
n
u
X
j
n
u
X
j
n
u
jX
E
e
E i
i
i
n
u
jXi
(10-49)
,
1
2
1
)
( 2
/
3
2
2 n
Y
n
o
n
u
u
(10-50)
2 2
/ 2
lim ( ) ,
u
Y
n
u e
(10-51)
PILLAI
22. 22
.
1
lim x
n
n
e
n
x
(10-52)
[Note that terms in (10-50) decay faster than
But (10-51) represents the characteristic function of a zero
mean normal r.v with variance and (10-45) follows.
The central limit theorem states that a large sum of
independent random variables each with finite variance
tends to behave like a normal random variable. Thus the
individual p.d.fs become unimportant to analyze the
collective sum behavior. If we model the noise phenomenon
as the sum of a large number of independent random
variables (eg: electron motion in resistor components), then
this theorem allows us to conclude that noise behaves like a
Gaussian r.v.
3/ 2
(1/ )
o n 3/ 2
1/ ].
n
2
since
PILLAI
23. 23
It may be remarked that the finite variance assumption is
necessary for the theorem to hold good. To prove its
importance, consider the r.vs to be Cauchy distributed, and
let
where each Then since
substituting this into (10-48), we get
which shows that Y is still Cauchy with parameter
In other words, central limit theorem doesn’t hold good for
a set of Cauchy r.vs as their variances are undefined.
.
2
1
n
X
X
X
Y n
(10-53)
).
(
C
Xi
,
)
( |
|u
X e
u
i
(10-54)
n
| |/
1
( ) ( / ) ~ ( ),
n
u n
Y X
i
u u n e C n
(10-55)
.
n
PILLAI
24. 24
Joint characteristic functions are useful in determining the
p.d.f of linear combinations of r.vs. For example, with X and
Y as independent Poisson r.vs with parameters and
respectively, let
Then
But from (6-33)
so that
i.e., sum of independent Poisson r.vs is also a Poisson
random variable.
.
Y
X
Z
(10-56)
).
(
)
(
)
( u
u
u Y
X
Z
(10-57)
)
1
(
)
1
( 2
1
)
(
,
)
(
ju
ju
e
Y
e
X e
u
e
u
(10-58)
)
(
)
( 2
1
)
1
)(
( 2
1
P
e
u
ju
e
Z
(10-59)
1
2
PILLAI